Difference between revisions of "File:Nembrant.jpg"

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{{oq|Nembrant.jpg|Original file ‎(361 × 871 pixels, file size: 65 KB, MIME type: image/jpeg)}}
⚫
[[Parametric plot]] of the branchpoint of function [[ArqNem]]$_q$,
  
⚫
The imaginary part of the branch point versus the real part for positive values of parameter $q$:
  
   
  ⚫
[[Parametric plot]] of the function [[NemBran]].
⚫
$x=\Re(\mathrm{NemBran}(q))$
  
  +
It appears as Figure 13.4 at page 162 of book
  +
«[[Superfunctions]]» <ref name="book">
  +
https://www.amazon.co.jp/-/en/Dmitrii-Kouznetsov/dp/6202672862 <br>
  +
https://www.morebooks.de/shop-ui/shop/product/978-620-2-67286-3 <br>
  +
https://mizugadro.mydns.jp/BOOK/468.pdf <br>
  +
Dmitrii Kouznetsov. [[Superfunctions]]. [[Lambert Academic Publishing]], 2020.
  +
</ref>.
   
  +
[[NemBran]] expresses the [[branchpoint]] of function [[ArqNem]]\(_q\),
−
$y=\Im(\mathrm{NemBran}(q))$
  
  ⚫
The imaginary part of the branch point versus the real part for positive values of parameter \(q\):
   
−
where $z=\mathrm{NemBran}(q)$ is branchpoint of function
 +
\(x=\Re(\mathrm{NemBran}(q))\)
  +
⚫
$z\mapsto\mathrm{ArqNem}_q(z)$
  
  ⚫
\(y=\Im(\mathrm{NemBran}(q))\)
  +
  +
where \(z=\mathrm{NemBran}(q)\) is branchpoint of function
  ⚫
\(z\mapsto\mathrm{ArqNem}_q(z)\)
   
 
Function [[NemBran]] is expressed through function [[NemBra]],
  
Function [[NemBran]] is expressed through function [[NemBra]],
−
that returns solution $z=\mathrm{NemBra}(q)$ of equation
 +
that returns solution \(z=\mathrm{NemBra}(q)\) of equation
   
−
$\mathrm{Nem}_q^{\,\prime}(z)=0$
 +
\(\mathrm{Nem}_q^{\,\prime}(z)=0\)
   
 
The representation has the following form:
  
The representation has the following form:
   
−
$\mathrm{NemBran}(q)=\mathrm{Nem}_q(\mathrm{NemBra}_q(z))$
 +
\(\mathrm{NemBran}(q)=\mathrm{Nem}_q(\mathrm{NemBra}_q(z))\)
   
−
While the moguls of the brachpoint is smaller than unity, and the linear term in the representation od the [[Nemtsov Function]] dominates. For this reason, the parametric plot of function [[NemBran]] looks similar to that of function [[NemBra]]. I remind the [[Nemtsov Function]] with parameter $q$ is defiend with
 +
While the moguls of the brachpoint is smaller than unity, and the linear term in the representation of the [[Nemtsov function]] dominates. For this reason, the parametric plot of function [[NemBran]] looks similar to that of function [[NemBra]]. I remind the [[Nemtsov function]] with parameter \(q\) is defined with
   
−
$\mathrm{Nem}_q(z)=z+z^3+q z^4$
 +
\(\mathrm{Nem}_q(z)=z+z^3+q z^4\)
   
−
Position of the branch point returned with function [[NemBra]] is the same for all the three versions of the inverse of the [[Nemtsov Function]]; not only for [[ArqNem]], but also for [[ArcNem]] and [[ArkNem]].
 +
Position of the branch point returned with function [[NemBra]] is the same for all the three versions of the inverse of the [[Nemtsov function]]; not only for [[ArqNem]], but also for [[ArcNem]] and [[ArkNem]].
   
 
==Usage==
  
==Usage==
   
−
Function [[NemBran]] is necessary for evaluation of [[ArqNem]]; this function, in its turn, is necessary for the extension of the [[Abel function]] of the [[Nemtsov function]], denoted with [[AuNem]], to the complex plane.
 +
Function [[NemBran]] is necessary for evaluation of [[ArqNem]]; this function, in its turn, is necessary for expression of the [[Abel function]] of the [[Nemtsov function]], denoted with [[AuNem]], in the complex plane.
⚫
Function [[AuNem]] should be described in the updated version of the book [[Superfunction]] in the example of [[exotic iterate]] of a transfer function $T$ such that
 
⚫
$T(0)\!=\!0$
  
⚫
$T'(0)\!=\!1$
  
⚫
$T''(0)\!=\!0$.
  
   
  ⚫
Function [[AuNem]] is described in the updated version of the book [[Superfunctions]] as the example of [[exotic iterate]] of a transfer function \(T\) such that
⚫
In such a way, the image Nembrant.jpg is prepared for the Book [[Superfunctions]].
  
  ⚫
\(T(0)\!=\!0\)
  ⚫
\(T'(0)\!=\!1\)
  ⚫
\(T''(0)\!=\!0\).
   
  ⚫
In such a way, the image Nembrant.jpg had been prepared for the Book [[Superfunctions]] <ref name="book">
⚫
One example of such function is $T=\sin$, bit function [[sin]] is symmetric, $\sin(-z)=-\sin(z)$.
  
  +
https://www.amazon.co.jp/-/en/Dmitrii-Kouznetsov/dp/6202672862 <br>
  +
https://www.morebooks.de/shop-ui/shop/product/978-620-2-67286-3 <br>
  +
https://mizugadro.mydns.jp/BOOK/468.pdf <br>
  +
Dmitrii Kouznetsov. [[Superfunctions]]. [[Lambert Academic Publishing]], 2020.
  +
</ref>.
  +
  ⚫
One example of such function is \(T=\sin\), bit function [[sin]] is symmetric, \(\sin(-z)=-\sin(z)\).
 
[[Superfunction]] and [[Abel Function]] of [[sin]] are denoted with [[SuSin]] and [[AuSin]].....
  
[[Superfunction]] and [[Abel Function]] of [[sin]] are denoted with [[SuSin]] and [[AuSin]].....
−
The corresponding iterates of [[sin]] are reported in 2014,
+
The corresponding iterates of [[sin]] are reported in 2014 at the [[Far East Jourmal of Mathematical Science]]
 
<ref>
  
<ref>
 
http://www.pphmj.com/references/8246.htm <br>
  
http://www.pphmj.com/references/8246.htm <br>
−
http://mizugadro.mydns.jp/PAPERS/2014susin.pdf D.Kouznetsov. Super sin. Far East Jourmal of Mathematical Science, v.85, No.2, 2014, pages 219-238.
 +
http://mizugadro.mydns.jp/PAPERS/2014susin.pdf D.Kouznetsov. Super sin. [[Far East Jourmal of Mathematical Science]], v.85, No.2, 2014, pages 219-238.
−
</ref>
 +
</ref>.
  +
  +
There was interesting to testy the algorithm for some function \(T\)
  +
that allows the same [[exotic iterate]] but have no symmetry \(T(-z)=-T(z)\).
  +
The simplest function of sich a kind is polynomial of 4 order.
  +
This polynomial is denoted with term \( \mathrm{Nem}_q \),
  +
it is defined at the top.
  +
In [[TORI]], this polinomial is called [[Nemtsov function]].
   
 
The example with [[Nemtsov function]] is important to show, that the symmetry is not essential for the construction of [[exotic iterate]]s; they can be constructed for asymmetric functions too.
  
The example with [[Nemtsov function]] is important to show, that the symmetry is not essential for the construction of [[exotic iterate]]s; they can be constructed for asymmetric functions too.
   
  +
Function [[Nembran]] is elementary function; it can be expressed in a closed form. However, this expession is not robust at very small nor at very large values of the argument. For this values,
⚫
==References==
  
  +
in the code below the two truncated asymptotic expansions are used:
−
<references/>
  
  +
Nembra0 and Nembrao.
   
 
==[[C++]] generator of the plot==
  
==[[C++]] generator of the plot==
   
  +
<pre>
−
<poem><nomathjax><nowiki>
  
 
#include <stdio.h>
  
#include <stdio.h>
 
#include <math.h>
  
#include <math.h>
Line 150: Line 175:
 
getchar();
  
getchar();
 
return 0;}
  
return 0;}
  +
</pre>
−
</nowiki></nomathjax></poem>
  
   
 
==[[Latex]] generator of labels==
  
==[[Latex]] generator of labels==
   
  +
<pre>
−
<poem><nomathjax><nowiki>
  
 
\documentclass[12pt]{article}
  
\documentclass[12pt]{article}
 
\usepackage{geometry}
  
\usepackage{geometry}
Line 203: Line 228:
 
\end{picture}
  
\end{picture}
 
\end{document}
  
\end{document}
  +
</pre>
−
</nowiki></nomathjax></poem>
  
  ⚫
==References==
  +
{{ref}}
   
  +
{{fer}}
  +
  +
==Keywords==
  +
«[[ArqNem]]»,
  +
«[[AuNem]]»,
  +
«[[Book]]»,
  +
«[[Branch point]]»,
  +
«[[C++]]»,
  +
«[[Latex]]»,
  +
«[[NemBra]]»,
  +
<b>«[[NemBran]]»</b>,
  +
<b>«[[Nemtsov function]]»</b>,
  +
«[[SuNem]]»,
  +
«[[AuNem]]»,
  +
«[[Parametric plot]]»,
  +
  ⚫
[[Category:ArqNem]]
  +
[[Category:AuNem]]
 
[[Category:Book]]
  
[[Category:Book]]
 
[[Category:BookPlot]]
  
[[Category:BookPlot]]
  +
[[Category:Branch point]]
 
[[Category:C++]]
  
[[Category:C++]]
 
[[Category:Latex]]
  
[[Category:Latex]]
−
[[Category:NemBrant]]
 +
[[Category:NemBra]]
−
[[Category:ArqNem]]
 +
[[Category:NemBran]]
 
[[Category:SuNem]]
  
[[Category:SuNem]]
⚫
[[Category:AuNem]]
  
 
[[Category:Nemtsov function]]
  
[[Category:Nemtsov function]]
 
[[Category:Parametric plot]]
  
[[Category:Parametric plot]]

Revision as of 06:19, 5 December 2025


Parametric plot of the function NemBran. It appears as Figure 13.4 at page 162 of book «Superfunctions» [1].

NemBran expresses the branchpoint of function ArqNem\(_q\), The imaginary part of the branch point versus the real part for positive values of parameter \(q\):

\(x=\Re(\mathrm{NemBran}(q))\)

\(y=\Im(\mathrm{NemBran}(q))\)

where \(z=\mathrm{NemBran}(q)\) is branchpoint of function \(z\mapsto\mathrm{ArqNem}_q(z)\)

Function NemBran is expressed through function NemBra, that returns solution \(z=\mathrm{NemBra}(q)\) of equation

\(\mathrm{Nem}_q^{\,\prime}(z)=0\)

The representation has the following form:

\(\mathrm{NemBran}(q)=\mathrm{Nem}_q(\mathrm{NemBra}_q(z))\)

While the moguls of the brachpoint is smaller than unity, and the linear term in the representation of the Nemtsov function dominates. For this reason, the parametric plot of function NemBran looks similar to that of function NemBra. I remind the Nemtsov function with parameter \(q\) is defined with

\(\mathrm{Nem}_q(z)=z+z^3+q z^4\)

Position of the branch point returned with function NemBra is the same for all the three versions of the inverse of the Nemtsov function; not only for ArqNem, but also for ArcNem and ArkNem.

Usage

Function NemBran is necessary for evaluation of ArqNem; this function, in its turn, is necessary for expression of the Abel function of the Nemtsov function, denoted with AuNem, in the complex plane.

Function AuNem is described in the updated version of the book Superfunctions as the example of exotic iterate of a transfer function \(T\) such that \(T(0)\!=\!0\) \(T'(0)\!=\!1\) \(T''(0)\!=\!0\).

In such a way, the image Nembrant.jpg had been prepared for the Book Superfunctions [1].

One example of such function is \(T=\sin\), bit function sin is symmetric, \(\sin(-z)=-\sin(z)\). Superfunction and Abel Function of sin are denoted with SuSin and AuSin..... The corresponding iterates of sin are reported in 2014 at the Far East Jourmal of Mathematical Science [2].

There was interesting to testy the algorithm for some function \(T\) that allows the same exotic iterate but have no symmetry \(T(-z)=-T(z)\). The simplest function of sich a kind is polynomial of 4 order. This polynomial is denoted with term \( \mathrm{Nem}_q \), it is defined at the top. In TORI, this polinomial is called Nemtsov function.

The example with Nemtsov function is important to show, that the symmetry is not essential for the construction of exotic iterates; they can be constructed for asymmetric functions too.

Function Nembran is elementary function; it can be expressed in a closed form. However, this expession is not robust at very small nor at very large values of the argument. For this values, in the code below the two truncated asymptotic expansions are used: Nembra0 and Nembrao.

C++ generator of the plot

#include <stdio.h>
#include <math.h>
#include <stdlib.h>
#include <complex>
typedef std::complex<double> z_type;
#define Re(x) x.real()
#define Im(x) x.imag()
#define I z_type(0.,1.)
#define DB double
#define DO(x,y) for(x=0;x<y;x++)

void ado(FILE *O, int X, int Y)
{       fprintf(O,"%c!PS-Adobe-2.0 EPSF-2.0\n",'%');
        fprintf(O,"%c%cBoundingBox: 0 0 %d %d\n",'%','%',X,Y);
        fprintf(O,"/M {moveto} bind def\n");
        fprintf(O,"/L {lineto} bind def\n");
        fprintf(O,"/S {stroke} bind def\n");
        fprintf(O,"/s {show newpath} bind def\n");
        fprintf(O,"/C {closepath} bind def\n");
        fprintf(O,"/F {fill} bind def\n");
        fprintf(O,"/o {.002 0 360 arc C S} bind def\n");
        fprintf(O,"/times-Roman findfont 20 scalefont setfont\n");
        fprintf(O,"/W {setlinewidth} bind def\n");
        fprintf(O,"/RGB {setrgbcolor} bind def\n");}
//#include "ado.cin"

z_type nem(DB q,z_type z){ return z*(1.+z*z*(1.+z*q)); }
z_type nem1(DB q,z_type z){ return 1.+z*z*(3.+z*(4.*q)); }
z_type nembra0(DB q){
return -0.5773502691896258*I+ 
        q*(2./9+
        q*(0.2138334330331947*I+
        q*(-0.2633744855967078 +
        q*(-0.3658927631901332*I+
        q*(0.5462581923487273 +
        q*(0.8556857213229570*I+
        q*(-1.387322393266609
        ))))))) ;}

z_type nembrao(DB q){ z_type x,y,z,s; x=conj(pow(z_type(-.25/q,0.),1./3.)); y=x*x; z=y*y;
s=1.+y*(1.+y*(1.+y*(2./3.+z*(-2./3.+y*(-7./9.+z*(11./9.+y*(130./91.)
       )     )     )        )         )         );
return s*x;}

z_type nembra(DB q){
if(fabs(q)<.021) return nembra0(q);
if(fabs(q) >20.) return nembrao(q);
z_type Q,v,V; Q=q*q; v=-1.-8.*Q+4.*sqrt(Q+4.*Q*Q); V=pow(v,1./3.); return (.25/q)*(-1.+1./V+V); }

z_type nembran(DB q){ return nem(q,nembra(q));}

int main(){ int m,n; z_type z,c,d,e; DB x,y,q,u,v;

DO(n,41){ q=-.04+.01*n; c=nembra0(q); d=nem(q,c); e=nem1(q,c); 
printf("%6.3lf %9.5lf %8.5lf %9.5lf %8.5lf %19.15lf %18.15lf\n",q,Re(c),Im(c), Re(d),Im(d), Re(e),Im(e) );}
printf("\n");
DO(n,41){ q=.1*n; c=nembra(q); d=nem(q,c); e=nem1(q,c); 
printf("%6.3lf %9.5lf %8.5lf %9.5lf %8.5lf %19.15lf %18.15lf\n",q,Re(c),Im(c), Re(d),Im(d), Re(e),Im(e) );}
printf("\n");
DO(n,41){ q=1.+1.*n; c=nembrao(q); d=nem(q,c); e=nem1(q,c); 
printf("%6.3lf %9.5lf %8.5lf %9.5lf %8.5lf %19.15lf %18.15lf\n",q,Re(c),Im(c), Re(d),Im(d), Re(e),Im(e) );}

//FILE *o; o=fopen("nembraplo.eps","w"); ado(o,154,604);
FILE *o; o=fopen("nembran.eps","w"); ado(o,154,404);
fprintf(o,"2 2 translate 1000 1000 scale 2 setlinecap\n");
#define M(x,y) fprintf(o,"%8.6f %8.6f M\n",(0.+x),(0.+y));
#define L(x,y) fprintf(o,"%8.6f %8.6f L\n",(0.+x),(0.+y));
#define o(x,y) fprintf(o,"%8.6f %8.6f o\n",(0.+x),(0.+y));

for(n=0;n<41;n+=5){ M(0,.01*n) L(.15,.01*n) }
for(n=0;n<31;n+=5){ M(.01*n,0) L(.01*n,.4) }
fprintf(o,".0006 W S 1 setlinejoin\n");
fprintf(o,"1 0 0 RGB .001 W\n");

for(n= 1;n<10;n++){ q=.1*n; c=nembran(q); x=Re(c); y=-Im(c); o(x,y); }
//for(n=11;n<20;n++){ q=.1*n; c=nembran(q); x=Re(c); y=-Im(c); o(x,y); }
//for(n=21;n<30;n++){ q=.1*n; c=nembran(q); x=Re(c); y=-Im(c); o(x,y); }
fprintf(o,"0 .5 0 RGB\n");
for(n= 1;n<10;n++){ q=1.*n; c=nembran(q); x=Re(c); y=-Im(c); o(x,y); }
fprintf(o,"0 0 1 RGB\n");
for(n= 1;n<10;n++){ q=10.*n; c=nembran(q); x=Re(c); y=-Im(c); o(x,y); }
fprintf(o,".5 0 .8 RGB\n");
for(n= 1;n<10;n++){ q=100.*n; c=nembran(q); x=Re(c); y=-Im(c); o(x,y); }
fprintf(o,"0 0 0 RGB\n");
for(n= 0;n<11;n+=10){ q=100.*n; c=nembran(q); x=Re(c); y=-Im(c); o(x,y); }
                      q=10000.; c=nembran(q); x=Re(c); y=-Im(c); o(x,y); 

for(n= 0;n<400;n++){ q=-.006+.04*(n+.01*n*n+.001*n*n*n); c=nembran(q); x=Re(c); y=-Im(c); if(n==0) M(x,y) else L(x,y); 
printf("%9.3lf %16.6lf %16.6lf\n",q,x,y);
}
L(0,0)
fprintf(o,"0 0 0 RGB .001 W 0 setlinecap S\n");

fprintf(o,"showpage\n%c%cTrailer\n",'%','%');
fclose(o);

system("epstopdf nembran.eps");
system("open     nembran.pdf");
getchar();
return 0;}

Latex generator of labels

\documentclass[12pt]{article}
\usepackage{geometry}
\paperwidth 174pt
\paperheight  420pt
\topmargin -100pt
\oddsidemargin -72pt
\textheight 800px
\parindent 0pt
\usepackage{graphicx}
\usepackage{rotating}
\newcommand \ing {\includegraphics}
\newcommand \sx {\scalebox}
\begin{document}
\begin{picture}(170,410)
%\put(20,14){\ing{nembraplo}}
\put(20,14){\ing{nembran}}
\put(8,411){\sx{1.4}{$y$}}
%\put(0,512){\sx{1.4}{$0.5$}}
%\put(0,412){\sx{1.4}{$0.4$}}
\put(0,312){\sx{1.4}{$0.3$}}
\put(0,212){\sx{1.4}{$0.2$}}
\put(0,112){\sx{1.4}{$0.1$}}
\put(8,12){\sx{1.4}{$0$}}
\put(20,1){\sx{1.4}{$0$}}
\put(60,1){\sx{1.4}{$0.05$}}
\put(112,1){\sx{1.4}{$0.1$}}
\put(162,2){\sx{1.4}{$x$}}
%\put(18,601){\sx{1.1}{$q\!=\!0$}}
\put(0,389){\sx{1.1}{$q\!=\!0$}}
%\put(63,593){\sx{1.1}{$q\!=\!0.2$}}
\put(43,404){\sx{1.1}{$q\!=\!0.2$}}
%\put(97,574){\sx{1.1}{$q\!=\!0.4$}}
\put(62,394){\sx{1.1}{$q\!=\!0.4$}}
\put(78,384){\sx{1.1}{$q\!=\!0.6$}}
\put(88,372){\sx{1.1}{$q\!=\!0.8$}}
\put(97,356){\sx{1.1}{$q\!=\!1$}}
\put(116,312){\sx{1.1}{$q\!=\!2$}}
\put(120,284){\sx{1.1}{$q\!=\!3$}}
\put(122,263){\sx{1.1}{$q\!=\!4$}}
\put(122,250){\sx{1.1}{$q\!=\!5$}}
\put(112,200){\sx{1.1}{$q\!=\!10$}}
\put(104,164){\sx{1.1}{$q\!=\!20$}}
\put(94,145){\sx{1.1}{$q\!=\!30$}}
\put( 74,99){\sx{1.1}{$q\!=\!100$}}
\put( 49,52){\sx{1.1}{$q\!=\!1000$}}
\put( 36,32){\sx{1.1}{$q\!=\!10000$}}
\end{picture}
\end{document}

References

Keywords

«ArqNem», «AuNem», «Book», «Branch point», «C++», «Latex», «NemBra», «NemBran», «Nemtsov function», «SuNem», «AuNem», «Parametric plot»,

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